On Birds, Bushes, and the Value of a Dollar

Ever since Twitter implemented the ability to attach a poll to a tweet, one of the most enthusiastic users of that functionality has been poker’s perpetual fan favorite, Daniel Negreanu. Interacting with his fans and the poker community in general has always been a big part of Negreanu’s personal brand, and one of the main reasons that he’s as popular as he is. One way in which he engages with his Twitter audience is through his “QOTDs,” that is, “questions of the day,” which often include a multiple-choice poll. Sometimes, these questions seem intended to make a political point, but at other times, he displays a more earnest desire to figure out how people think; Negreanu takes a fairly psychological, rather than mathematical approach to poker, so it’s natural that he’d be interested in the workings of the human mind more generally.

Some of the Negreanu QOTD polls that I most enjoy have to do with how people value money. Specifically, how they value a certain amount of strings-free, immediate money, compared to a large amount of money with conditions attached. The most recent such question from him came last week, in the following tweet. The voting period has concluded, so you can see the results as well:

QOTD: if your net worth was $1 million would you risk it ALL on a coin flip where if you won you got $10 million?

The trouble with Twitter polls is usually that the character limits don’t allow for either the question or the responses to be phrased with a whole lot of precision. People end up making different assumptions about what’s being asked, which can affect the results. Sometimes, however, that ambiguity is actually what’s interesting about a question, because people’s answers tell you something about the assumptions they’re making, which in turn gives you some insight into their psychology.

Here, the key ambiguity lies in the initial situation presented: “If your net worth was $1 million…” Are we assuming ourselves to be in the position of a typical person with a net worth of $1 million (with the sort of upper middle-class job and lifestyle that entails), or ourselves, with money and possessions added or removed to adjust our net worth to $1 million, while leaving everything else about our situation unchanged? I think the results are easier to interpret and make more sense if we guess that most people are assuming the latter.

Where this is most relevant is when we think about the situation we’d find ourselves in if we take the coin flip and lose. After all, having $1 million or $10 million might allow or require lifestyle adjustments of one form or another depending on how one was living before, but no one is going to be in bad shape with that kind of money. Deprived of all assets and physical property, however, all sorts of aspects of one’s life become relevant: family, connections, job, credit rating, and so forth. For someone with ready access to short-term help and long-term income, being penniless is simply a matter of being “broke” in the short term. For others, however, losing everything would likely mean being destitute for life, or at least the foreseeable future.

The invisible safety net

In part, then, this question tests how much privilege one possesses, as a fairly decent though perhaps incomplete definition of privilege would be exactly this: the invisible safety net which surrounds everyone to varying degrees, and makes it easier (or harder, for those who lack privilege) to bounce back from a bad situation.

The funny thing about privilege is that the more of it someone has, the less aware of it they tend to be, unless deliberately looking for it. That’s precisely because it’s the sort of thing you only have to worry about if you don’t have it. If you’re highly privileged (whether by birth or circumstance), you’ll be more-or-less fine whatever happens, so you don’t have a whole lot of “what ifs” on your mind. Those less privileged, on the other hand, do have to face the possibility that a stroke of bad luck could put them in a position from which they can’t recover.

You can see this quite easily in some of the responses to the poll. Particularly telling is that of poker player, TV personality and model Liv Boeree. Boeree is one of the co-founders of the “effective altruism” charity Raising for Effective Giving (REG), so one would assume she’s sufficiently well-acquainted with social issues that she knows that her personal situation is not reflective of most people’s experience. Even so, she initially expressed bafflement that anyone would decline the coin flip, given that it’s so clearly profitable from a statistical perspective.

Once people started to respond to her and she’d had a chance to think about it, Boeree quickly realized what the question was really about and why so many people would take the sure million rather than risk the coin flip. This is exactly why I think the immediate gut reaction to the question is what’s important, however; it largely comes down to the situation that springs to mind when one tries to imagine “zero net worth.” For Boeree, that would presumably just mean crashing at a friend’s place for a few weeks and asking the bank for a loan while waiting for her next paycheck to come in, but not everyone has such options available.

Risk aversion and utility functions

That’s the sociological way of looking at it, anyway. An economist would say that what the question is measuring is something called risk aversion. Risk aversion is, in a nutshell, a measurement of the extent to which a given person experiences (or believes they will experience) diminishing returns from additional money. In other words, what you can do with $10 million doesn’t necessarily make your life ten times better than what you can do with $1 million, so there’s more to consider in this sort of question than just what is going to get you the most money on average.

The subjective desirability or practical use of a certain amount of money is called utility and how this changes for a given person as the amounts involved get larger is called their utility function. If that’s a straight line, then it means the person values each dollar equally no matter how many they already have, and they’re described as risk neutral. Most people, however, are risk averse, meaning that additional money has less utility to them – on a dollar by dollar basis – than the money they already have.

Tournament poker players are of course very familiar with a special (and unusually quantifiable) case of this phenomenon in the form of ICM (short for “independent chip model”). Since tournament payouts are based on finish position, chips don’t translate directly into real-world money; rather, the value of a player’s stack has to do with their likelihood of outlasting other players. Chips are therefore usually worth more when a player has fewer of them, since that’s when the player’s survival is most at risk.

What’s the going rate for birds these days, anyway?

So, while an economist would understand ICM as a concrete reason to be risk averse in a poker tournament when a pay jump is imminent, it’s equally possible for a poker player to apply the same logic in reverse, to see how people at a social disadvantage have “life ICM” to consider.

This is important to realize, because poker players often toss the term risk aversion around as if it were an irrational personality characteristic of people themselves, rather than a function of their overall situation. Of course, some people are also intrinsically more risk-averse than others, for reasons of personality – that is, two people can certainly have differently-shaped utility functions for the exact same situation – but extrinsic factors, like the ones we’ve already discussed, definitely exist as well.

In other words, there’s a lot of merit to the old saw about a bird in the hand being worth two in the bush, except that the two-to-one figure shouldn’t be taken too literally. A bird in the hand has more value to nearly anyone than an equivalent bird in the bush, yet the exact value ratio is going to depend both on the temperament of the bird-holder, and her life situation. Some people might be perfectly justified in taking the chance at a second bird, but others may well need a shot at the whole damn flock before they’re willing to let go of the sure thing.

Does this life have rebuys?

Since we’re applying poker experience to the theory of life, rather than the other way around, I want to finish by asking whether the poker tournament analogy has any insight to offer us in terms of why going broke is so much worse for some people than others, and thus why those people need to play the game of life in a more risk-averse manner. I think it does.

I think the relevant parallel here is with rebuy and unlimited re-entry tournaments. Professionals like these for a few reasons, but the main one is that it allows them to play in a risk-neutral manner, which is typically the most profitable way to play poker. Without the option to rebuy or re-enter a tournament, even deep-pocketed pros suffer from a form of risk aversion related to the opportunity cost of busting out. That is, that there usually won’t be an equivalent tournament running at the same time, and the cash games happening alongside might not be as lucrative for the player in question; thus, busting out doesn’t only lose the buy-in, but also the (presumably profitable) opportunity to play the tournament in the first place.

As long as the professional has both the option and the bankroll to fire multiple bullets in a tournament, this opportunity cost disappears, so his hands are free to take the most profitable spots he can find, without worrying about the risk of busting. As long as that option remains available, the pro’s effective stack is infinitely deep (though he’s limited in how much of it he can wager in a single hand), and the strategic considerations are more similar to those in a cash game where the value of chips is linear. This gives him an advantage over a recreational player or a less well-heeled regular who is more reluctant to bust out, and must therefore pass up some slightly profitable but high-variance spots.

The same is true of someone whose real-life situation allows them to “rebuy” by calling in favors from friends or business connections, or simply by going into debt while feeling relatively assured of being able to pay off that debt in short order. Part of the reason someone in a good life situation can afford to be less risk averse is that they hold a lot of social capital that is less tangible but no less real than what’s counted in their net worth.

Probably the most frequently asked risk aversion hypothetical in poker involves the situation of facing a preflop all-in shove on the first hand of the World Series of Poker Main Event. There are lots of variations on the question, but one example would be that in the first hand, you’re dealt pocket Queens in the big blind. It folds around to the small blind, who flashes you Ace-King, then goes all-in. You’re slightly better than 50/50 to win, but only slightly, and for many people, playing the Main Event is a once-in-a-lifetime kind of thing. What makes it a fun question to ask, then, is that it’s an obvious yes for some people and an obvious no for others, and people who answer one way or the other often assume everyone who answers the other way is crazy.

If you asked the same question, but for the first hand of a low stakes rebuy event, it’s a totally different story. There, almost everyone would go for it; even if they’ve never heard of terms like risk aversion, it’s intuitively obvious that losing your stack and rebuying is not a big deal, whereas missing your shot at the Main Event is. It’s exactly the same thing when it comes to life; some of us are playing a re-entry event, and others are playing a freezeout. It’s not surprising, then, that some of us feel like you’d be crazy to take Daniel’s hypothetical coinflip, and others feel just as strongly that you’d be crazy not to.

Alex Weldon (@benefactumgames) is a freelance writer, game designer and semipro poker player from Montreal, Quebec, Canada.

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